In this work, we first focus on the mathematical structure of the three-dimensional (3D) Ising model. In the Clifford algebraic representation, many internal factors exist in the transfer matrices of the 3D Ising model, which are ascribed to the topology of the 3D space and the many-body interactions of spins. They result in the nonlocality, the nontrivial topological structure, as well as the long-range entanglement between spins in the 3D Ising model. We review briefly the exact solution of the ferromagnetic 3D Ising model at the zero magnetic field, which was derived in our previous work. Then, the framework of topological quantum statistical mechanics is established, with respect to the mathematical aspects (topology, algebra, and geometry) and physical features (the contribution of topology to physics, Jordan-von Neumann-Wigner framework, time average, ensemble average, and quantum mechanical average). This is accomplished by generalizations of our findings and observations in the 3D Ising models. Finally, the results are generalized to topological quantum field theories, in consideration of relationships between quantum statistical mechanics and quantum field theories. It is found that these theories must be set up within the Jordan-von Neumann-Wigner framework, and the ergodic hypothesis is violated at the finite temperature. It is necessary to account the time average of the ensemble average and the quantum mechanical average in the topological quantum statistical mechanics and to introduce the parameter space of complex time (and complex temperature) in the topological quantum field theories. We find that a topological phase transition occurs near the infinite temperature (or the zero temperature) in models in the topological quantum statistical mechanics and the topological quantum field theories, which visualizes a symmetrical breaking of time inverse symmetry.

We provide the following result and its discrete equivalent: Let $f \colon I^n \to \mathbb{R}^{n-1}$ be a continuous function. Then, there exist a point $p \in \mathbb{R}^{n-1}$ and a compact subset $S \subset f^{-1}\left[\left\{p\right\}\right]$ which connects some opposite faces of the $n$-dimensional unit cube $I^n$. We give an example that shows it cannot be generalized to path-connected sets. Additionally, we show that a version of the Steinhaus Chessboard Theorem and the Brouwer Fixed Point Theorem are simple consequences of this result.

This paper presents reconfigurable intelligent surface (RIS)-aided deep learning (DL)-based spectrum sensing for next-generation cognitive radios. To that end, the secondary user (SU) monitors the primary transmitter (PT) signal, where the RIS plays a pivotal role in increasing the strength of the PT signal at the SU. The spectrograms of the synthesized dataset, including the 4G LTE and 5G NR signals, are mapped to images utilized for training the state-of-art object detection approaches, namely Detectron2 and YOLOv7. By conducting extensive experiments using a real RIS prototype, we demonstrate that the RIS can consistently and significantly improve the performance of the DL detectors to identify the PT signal type along with its time and frequency utilization. This study also paves the way for optimizing spectrum utilization through RIS-assisted CR application in next-generation wireless communication systems.

We derive the standard power-type NLS as a scaling limit of the Gabitov--Turitsyn dispersion-managed NLS, using the $2d$ defocusing, cubic equation as a model case. In particular, we obtain global-in-time scattering solutions to the dispersion-managed NLS for large scale data of arbitrary $L^2$-norm.

The relation between negatively curved spaces and their boundaries is important for various rigidity problems. In \cite{biswas2024quasi}, the class of Gromov hyperbolic spaces called maximal Gromov hyperbolic spaces was introduced, and the boundary functor $X \mapsto \partial X$ was shown to give an equivalence of categories between maximal Gromov hyperbolic spaces (with morphisms being isometries) and a class of compact quasi-metric spaces called quasi-metric antipodal spaces (with morphisms being Moebius homeomorphisms). The proof of this equivalence involved the construction of a filling functor $Z \mapsto {\mathcal M}(Z)$, associating to any quasi-metric antipodal space $Z$ a maximal Gromov hyperbolic space ${\mathcal M}(Z)$. We study the ``continuity" properties of the boundary and filling functors. We show that convergence of a sequence of quasi-metric antipodal spaces (in a certain sense called ``almost-isometric convergence") implies convergence (in the Gromov-Hausdorff sense) of the associated maximal Gromov hyperbolic spaces. Conversely, we show that convergence of maximal Gromov hyperbolic spaces together with a natural hypothesis of ``equicontinuity" on the boundaries implies convergence of boundaries. We use this to show that Gromov-Hausdorff convergence of a sequence of proper, geodesically complete CAT(-1) spaces implies Gromov-Hausdorff convergence of their boundaries equipped with visual metrics. We also show that convergence of maximal Gromov hyperbolic spaces to a maximal Gromov hyperbolic space with finite boundary implies convergence of boundaries.

The exact solution of ferromagnetic two-dimensional (2D) Ising model with a transverse field, which can be used to describe the critical phenomena in low-dimensional quantum spin systems, is derived by equivalence between the ferromagnetic 2D Ising model with a transverse field and the ferromagnetic three-dimensional (3D) Ising model. The results obtained in this work can be extended to be suitable for the antiferromagnetic 2D Ising model with a transverse field in the cases without frustration.

We establish a small-data modified scattering result for the $1d$ cubic dispersion-managed NLS (with time-dependent dispersion map) for initial data in a weighted space.

Individuals often act without knowing the full structure of the social network in which they are or will be embedded. We study how an individual's beliefs about their networks shapes their behavior when actions are peer interactive. Agents use what they know about the network to forecast their peers' actions. Those peers' actions depend on their beliefs, which then generate an iterative expression what we call "Iterative Belief Centrality." Agents' beliefs formed based on what they each see of the network are heterogeneous, depend on their network position, and can be correlated across connected agents. The resulting equilibrium behavior nests both complete-information and degree-based models as special cases, but more generally can differ systematically. If people's beliefs about the network satisfy a natural monotonicity condition in how connected they are, then belief iteration (fully rationally) amplifies behavioral differences across the network, increasing actions of more-connected and decreasing actions of less-connected agents relative to situations with homogeneous beliefs. We also show how positive correlation in people's positions in the network even further amplifies the variance of behavior. The framework provides a unified and tractable theory of network-based behavior with implications for many applications.

We show that a compact complex parallelisable nilmanifold has unobstructed deformations if and only if its associated Lie algebra satisfies a reality condition and is a free Lie algebra in a variety of Lie algebras, that is, defined by a verbal ideal in a free Lie algebra. We provide a partial classification of verbal ideals and show that there are finitely many such Lie algebras up to dimension 19, whereas infinite families start to appear in dimension 20. As a consequence, there are finitely many complex homotopy types of unobstructed complex parallelisable nilmanifolds up to dimension 19, and infinitely many in dimension 20.

We consider Riemann's Xi function $ξ(s)$ which is evaluated at $s = \frac{1}{2} + σ+ i ω$, given by $ξ(\frac{1}{2} + σ+ i ω)= E_{pω}(ω)$, where $σ, ω$ are real and compute its inverse Fourier transform given by $E_p(t)$. We study the properties of $E_p(t)$ and a promising new method is presented which could be used to show that the Fourier Transform of $E_p(t)$ given by $E_{pω}(ω) = ξ(\frac{1}{2} + σ+ i ω)$ does not have zeros for finite and real $ω$ when $0 < |σ| < \frac{1}{2}$, corresponding to the critical strip excluding the critical line.

We prove several scattering results for dispersion-managed nonlinear Schrödinger equations. In particular, we establish small-data scattering for both `intercritical' and `mass-subcritical' powers by suitable modifications of the standard approach via Strichartz estimates. In addition, we prove scattering for arbitrary data in a weighted Sobolev space for intercritical powers by establishing a pseudoconformal energy estimate. We also rule out (unmodified) scattering for sufficiently low powers. Finally, we give some remarks concerning blowup for the focusing equation.

In this note, we extend the notion of minimal gaps to the higher dimensional sequences. We bound the minimal gap for $(\{\boldsymbol{a}_n\boldsymbolα\}),$ $(\{a_n\boldsymbolα\})$ and $(\{\boldsymbol{a}_n\cdot\boldsymbolα\})$ in terms of cardinality of the difference set of $a_n$ and $\boldsymbol{a}_n,$ where $a_n$ and $a_n^{(1)},\dots,a_n^{(d)}$ are sequences of distinct integers.

As Large Language Models (LLMs) have become integral to both research and daily operations, rigorous evaluation is crucial. This assessment is important not only for individual tasks but also for understanding their societal impact and potential risks. Despite extensive efforts to examine LLMs from various perspectives, there is a noticeable lack of multi-agent AI models specifically designed to evaluate the performance of different LLMs. To address this gap, we introduce a novel multi-agent AI model that aims to assess and compare the performance of various LLMs. Our model consists of eight distinct AI agents, each responsible for retrieving code based on a common description from different advanced language models, including GPT-3.5, GPT-3.5 Turbo, GPT-4, GPT-4 Turbo, Google Bard, LLAMA, and Hugging Face. Our developed model utilizes the API of each language model to retrieve code for a given high-level description. Additionally, we developed a verification agent, tasked with the critical role of evaluating the code generated by its counterparts. We integrate the HumanEval benchmark into our verification agent to assess the generated code's performance, providing insights into their respective capabilities and efficiencies. Our initial results indicate that the GPT-3.5 Turbo model's performance is comparatively better than the other models. This preliminary analysis serves as a benchmark, comparing their performances side by side. Our future goal is to enhance the evaluation process by incorporating the Massively Multitask Benchmark for Python (MBPP) benchmark, which is expected to further refine our assessment. Additionally, we plan to share our developed model with twenty practitioners from various backgrounds to test our model and collect their feedback for further improvement.

Reconfigurable intelligent surface (RIS)-empowered communications represent exciting prospects as one of the promising technologies capable of meeting the requirements of the sixth generation networks such as low-latency, reliability, and dense connectivity. However, validation of test cases and real-world experiments of RISs are imperative to their practical viability. To this end, this paper presents a physical demonstration of an RIS-assisted communication system in an indoor environment in order to enhance the coverage by increasing the received signal power. We first analyze the performance of the RIS-assisted system for a set of different locations of the receiver and observe around 10 dB improvement in the received signal power by careful RIS phase adjustments. Then, we employ an efficient codebook design for RIS configurations to adjust the RIS states on the move without feedback channels. We also investigate the impact of an efficient grouping of RIS elements, whose objective is to reduce the training time needed to find the optimal RIS configuration. In our extensive experimental measurements, we demonstrate that with the proposed grouping scheme, training time is reduced from one-half to one-eighth by sacrificing only a few dBs in received signal power.

We report quantum oscillation measurements on the kagome compounds ATi$_3$Bi$_5$ (A=Rb, Cs) in magnetic fields up to 41.5 T and temperatures down to 350 mK. In addition to the frequencies observed in previous studies, we have observed multiple unreported frequencies above 2000 T in CsTi$_3$Bi$_5$ using a tunnel diode oscillator technique. We compare these results against density functional theory calculations and find good agreement with the calculations in the number of peaks observed, frequency, and the dimensionality of the Fermi surface. For RbTi$_3$Bi$_5$ we have obtained a different quantum oscillation spectrum, although calculated quantum oscillation frequencies for the Rb compound are remarkably similar to the Cs compound, calling for further studies.

There have been recently many studies demonstrating that the performance of wireless communication systems can be significantly improved by a reconfigurable intelligent surface (RIS), which is an attractive technology due to its low power requirement and low complexity. This paper presents a measurement-based characterization of RISs for providing physical layer security, where the transmitter (Alice), the intended user (Bob), and the eavesdropper (Eve) are deployed in an indoor environment. Each user is equipped with a software-defined radio connected to a horn antenna. The phase shifts of reflecting elements are software controlled to collaboratively determine the amount of received signal power at the locations of Bob and Eve in such a way that the secrecy capacity is aimed to be maximized. An iterative method is utilized to configure a Greenerwave RIS prototype consisting of 76 passive reflecting elements. Computer simulation and measurement results demonstrate that an RIS can be an effective tool to significantly increase the secrecy capacity between Bob and Eve.

We introduce a new class of $(α,β)$-type exact solutions in Finsler gravity closely related to the well-known pp-waves in general relativity. Our class contains most of the exact solutions currently known in the literature as special cases. The linearized versions of these solutions may be interpretted as Finslerian gravitational waves, and we investigate the physical effect of such waves. More precisely, we compute the Finslerian correction to the radar distance along an nterferometer arm at the moment a Finslerian gravitational wave passes a detector. We come to the remarkable conclusion that the effect of a Finslerian gravitational wave on an interferometer is indistinguishable from that of standard gravitational wave in general relativity. Along the way we also physically motivate a modification of the Randers metric and prove that it has some very interesting properties.

A clear definition of system dynamics modeling can provide shared understanding and clarify the impact of the field. We introduce a set of characteristics that define quantitative system dynamics, selected to capture core philosophy, describe theoretical and practical principles, and apply to historical work but be flexible enough to remain relevant as the field progresses. The defining characteristics are: (1) models are based on causal feedback structure, (2) accumulations and delays are foundational, (3) models are equation-based, (4) concept of time is continuous, and (5) analysis focuses on feedback dynamics. We discuss the implications of these principles and use them to identify research opportunities in which the system dynamics field can advance. These research opportunities include causality, disaggregation, data science and artificial intelligence, and contributing to scientific advancement. Progress in these areas has the potential to improve both the science and practice of system dynamics.

Developing complex, reliable advanced accelerators requires a coordinated, extensible, and comprehensive approach in modeling, from source to the end of beam lifetime. We present highlights in Exascale Computing to scale accelerator modeling software to the requirements set for contemporary science drivers. In particular, we present the first laser-plasma modeling on an exaflop supercomputer using the US DOE Exascale Computing Project WarpX. Leveraging developments for Exascale, the new DOE SCIDAC-5 Consortium for Advanced Modeling of Particle Accelerators (CAMPA) will advance numerical algorithms and accelerate community modeling codes in a cohesive manner: from beam source, over energy boost, transport, injection, storage, to application or interaction. Such start-to-end modeling will enable the exploration of hybrid accelerators, with conventional and advanced elements, as the next step for advanced accelerator modeling. Following open community standards, we seed an open ecosystem of codes that can be readily combined with each other and machine learning frameworks. These will cover ultrafast to ultraprecise modeling for future hybrid accelerator design, even enabling virtual test stands and twins of accelerators that can be used in operations.

We prove bubble-tree convergence of sequences of gradient Ricci shrinkers with uniformly bounded entropy and uniform local energy bounds, refining the compactness theory of Haslhofer-Mueller. In particular, we show that no energy concentrates in neck regions, a result which implies a local energy identity for the sequence. Direct consequences of these results are an identity for the Euler characteristic and a local diffeomorphism finiteness theorem.

Given a Hermitian holomorphic vector bundle over a complex manifold, consider its flag bundles with the associated universal vector bundles endowed with the induced metrics. We prove that the universal formula for the push-forward of a polynomial in the Chern classes of all the possible universal vector bundles also holds pointwise at the level of Chern forms. A key step in our proof is the explicit computation, at a point of any flag bundle, of the Chern curvature of the universal vector bundles with the induced metrics. As an application, we provide an alternative version of the Jacobi-Trudi identity at the level of differential forms. We also show the positivity of a family of polynomials in the Chern forms of Griffiths semipositive vector bundles. This latter result partially confirms the Griffiths' conjecture on positive characteristic forms, which has raised considerable interest in recent years.

The global financial crisis of 2007-2009 highlighted the crucial role systemic risk plays in ensuring stability of financial markets. Accurate assessment of systemic risk would enable regulators to introduce suitable policies to mitigate the risk as well as allow individual institutions to monitor their vulnerability to market movements. One popular measure of systemic risk is the conditional value-at-risk (CoVaR), proposed in Adrian and Brunnermeier (2011). We develop a methodology to estimate CoVaR semi-parametrically within the framework of multivariate extreme value theory. According to its definition, CoVaR can be viewed as a high quantile of the conditional distribution of one institution's (or the financial system) potential loss, where the conditioning event corresponds to having large losses in the financial system (or the given financial institution). We relate this conditional distribution to the tail dependence function between the system and the institution, then use parametric modelling of the tail dependence function to address data sparsity in the joint tail regions. We prove consistency of the proposed estimator, and illustrate its performance via simulation studies and a real data example.

We develop a refined singularity analysis for the Ricci flow by investigating curvature blow-up rates locally. We first introduce general definitions of Type I and Type II singular points and show that these are indeed the only possible types of singular points. In particular, near any singular point the Riemannian curvature tensor has to blow up at least at a Type I rate, generalising a result of Enders, Topping and the first author that relied on a global Type I assumption. We also prove analogous results for the Ricci tensor, as well as a localised version of Sesum's result, namely that the Ricci curvature must blow up near every singular point of a Ricci flow, again at least at a Type I rate. Finally, we show some applications of the theory to Ricci flows with bounded scalar curvature.

We present some geometric applications, of global character, of the bubbling analysis developed by Buzano and Sharp for closed minimal surfaces, obtaining smooth multiplicity one convergence results under upper bounds on the Morse index and suitable lower bounds on either the genus or the area. For instance, we show that given any Riemannian metric of positive scalar curvature on the three-dimensional sphere the class of embedded minimal surfaces of index one and genus $γ$ is sequentially compact for any $γ\geq 1$. Furthemore, we give a quantitative description of how the genus drops as a sequence of minimal surfaces converges smoothly, with mutiplicity $m\geq 1$, away from finitely many points where curvature concentration may happen. This result exploits a sharp estimate on the multiplicity of convergence in terms of the number of ends of the bubbles that appear in the process.

In this article, we prove a general and rather flexible upper bound for the heat kernel of a weighted heat operator on a closed manifold evolving by an intrinsic geometric flow. The proof is based on logarithmic Sobolev inequalities and ultracontractivity estimates for the weighted operator along the flow, a method which was previously used by Davies in the case of a non-evolving manifold. This result directly implies Gaussian-type upper bounds for the heat kernel under certain bounds on the evolving distance function; in particular we find new proofs of Gaussian heat kernel bounds on manifolds evolving by Ricci flow with bounded curvature or positive Ricci curvature. We also obtain similar heat kernel bounds for a class of other geometric flows.

We develop a Clifford algebra approach for 3D Ising model. By utilizing some mathematical facts of the direct product of matrices and their trace, we expand the dimension of the transfer matrices V of the 3D Ising system by adding unit matrices I (with compensation of a factor) and adjusting their sequence, which do not change the trace of the transfer matrices V (Theorem I: Trace Invariance Theorem). It allows us to perform a linearization process on sub-transfer-matrices (Theorem II: Linearization Theorem). It is found that locally for each site j, the internal factor Wj in the transfer matrices can be treated as a boundary factor, which can be dealt with by a procedure similar to the Onsager-Kaufman approach for the boundary factor U in the 2D Ising model. This linearization process splits each sub-transfer matrix into 2n sub-spaces (and the whole system into 2nl sub-spaces). Furthermore, a local transformation is employed on each of the sub-transfer matrices (Theorem III: Local Transformation Theorem). The local transformation trivializes the non-trivial topological structure, while it generalizes the topological phases on the eigenvectors. This is induced by a gauge transformation in the Ising gauge lattice that is dual to the original 3D Ising model. The non-commutation of operators during the processes of linearization and local transformation can be dealt with to be commutative in the framework of the Jordan-von Neumann-Wigner procedure, in which the multiplication in Jordan algebras is applied instead of the usual matrix multiplication AB (Theorem IV: Commutation Theorem).

In Noor (2007)[Muhammad Aslam Noor, New iterative schemes for nonlinear equations, Appl. Math. Comput. 187 (2007) 937-943], proposed an algorithm namely \textbf{Algorithm 2.4} and established a proof to show cubic convergence. The presented proof is wrong. In this article, we show that \textbf{Algorithm 2.4} is not cubically convergent.

In the paper [Muhammad Aslam Noor, Khalida Inayat Noor, Three-step iterative methods for nonlinear equations, Applied Mathematics and Computation, 183 (2006), pp. 322-327 ], Authors presented an algorithm (\textbf{Algorithm 2.3}) and stated a theorem (\textbf{Theorem 2.3}) to prove the cubic order of convergence but the given proof does not show cubic order of convergence. Actually, the mathematical derivation steps to develop the \textbf{Algorithm 2.3} are wrong. In this note, we present the correct mathematical developments and finally provide computational order of convergence in the favor of our claim and provide the generalization of the method.

An overview of the mathematical structure of the three-dimensional (3D) Ising model is given, from the viewpoints of topologic, algebraic and geometric aspects. By analyzing the relations among transfer matrices of the 3D Ising model, Reidemeister moves in the knot theory, Yang-Baxter and tetrahedron equations, the following facts are illustrated for the 3D Ising model: 1) The complexified quaternion basis constructed for the 3D Ising model represents naturally the rotation in a (3 + 1) - dimensional space-time, as a relativistic quantum statistical mechanics model, which is consistent with the 4-fold integrand of the partition function by taking the time average. 2) A unitary transformation with a matrix being a spin representation in 2^(nlo)-space corresponds to a rotation in 2nlo-space, which serves to smooth all the crossings in the transfer matrices and contributes as the non-trivial topologic part of the partition function of the 3D Ising model. 3) A tetrahedron relation would ensure the commutativity of the transfer matrices and the integrability of the 3D Ising model, and its existence is guaranteed also by the Jordan algebra and the Jordan-von Neumann-Wigner procedures. 4) The unitary transformation for smoothing the crossings in the transfer matrices changes the wave functions by complex phases Φx, Φy, and Φz. The relation with quantum field and gauge theories, physical significance of weight factors are discussed in details. The conjectured exact solution is compared with numerical results, and singularities at/near infinite temperature are inspected. The analyticity in β = 1/(kB T) of both the hard-core and Ising models has been proved for β > 0, not for β = 0. Thus the high-temperature series cannot serve as a standard for judging a putative exact solution of the 3D Ising model.

In opportunistic networks the existence of a simultaneous path is not assumed to transmit a message between a sender and a receiver. Information about the context in which the users communicate is a key piece of knowledge to design efficient routing protocols in opportunistic networks. But this kind of information is not always available. When users are very isolated, context information cannot be distributed, and cannot be used for taking efficient routing decisions. In such cases, context oblivious based schemes are only way to enable communication between users. As soon as users become more social, context data spreads in the network, and context based routing becomes an efficient solution. In this paper we design an integrated routing protocol that is able to use context data as soon as it becomes available and falls back to dissemination based routing when context information is not available. Then, we provide a comparison between Epidemic and PROPHET, these are representative of context oblivious and context aware routing protocols. Our results show that integrated routing protocol is able to provide better result in term of message delivery probability and message delay in both cases when context information about users is available or not.